Statistics > Machine Learning

Abstract: We developed a new class of physics-informed generative adversarial networks
(PI-GANs) to solve in a unified manner forward, inverse and mixed stochastic
problems based on a limited number of scattered measurements. Unlike standard
GANs relying only on data for training, here we encoded into the architecture
of GANs the governing physical laws in the form of stochastic differential
equations (SDEs) using automatic differentiation. In particular, we applied
Wasserstein GANs with gradient penalty (WGAN-GP) for its enhanced stability
compared to vanilla GANs. We first tested WGAN-GP in approximating Gaussian
processes of different correlation lengths based on data realizations collected
from simultaneous reads at sparsely placed sensors. We obtained good
approximation of the generated stochastic processes to the target ones even for
a mismatch between the input noise dimensionality and the effective
dimensionality of the target stochastic processes. We also studied the
overfitting issue for both the discriminator and generator, and we found that
overfitting occurs also in the generator in addition to the discriminator as
previously reported. Subsequently, we considered the solution of elliptic SDEs
requiring approximations of three stochastic processes, namely the solution,
the forcing, and the diffusion coefficient. We used three generators for the
PI-GANs, two of them were feed forward deep neural networks (DNNs) while the
other one was the neural network induced by the SDE. Depending on the data, we
employed one or multiple feed forward DNNs as the discriminators in PI-GANs.
Here, we have demonstrated the accuracy and effectiveness of PI-GANs in solving
SDEs for up to 30 dimensions, but in principle, PI-GANs could tackle very high
dimensional problems given more sensor data with low-polynomial growth in
computational cost.